# `Yog.DisjointSet`
[🔗](https://github.com/code-shoily/yog_ex/blob/v0.97.0/lib/yog/disjoint_set.ex#L1)

Disjoint Set Union (Union-Find) data structure for efficient set operations.

The disjoint-set data structure maintains a partition of elements into disjoint (non-overlapping)
sets. It provides near-constant time operations to add elements, find which set an element
belongs to, and merge two sets together.

## Key Operations

| Operation | Function | Complexity |
|-----------|----------|------------|
| Make Set | `add/2` | O(1) |
| Find | `find/2` | O(α(n)) amortized |
| Union | `union/3` | O(α(n)) amortized |

Where α(n) is the [inverse Ackermann function](https://en.wikipedia.org/wiki/Ackermann_function#Inverse),
which grows so slowly that it is effectively a small constant (≤ 4) for all practical inputs.

## Optimizations

This implementation uses two key optimizations:
- **Path Compression**: Flattens the tree structure during find operations, making future queries faster
- **Union by Rank**: Attaches the shorter tree under the taller tree to minimize tree height

## Use Cases

- [Kruskal's MST algorithm](https://en.wikipedia.org/wiki/Kruskal%27s_algorithm) - detecting cycles
- Connected components in dynamic graphs
- Equivalence relations and partitioning
- Percolation theory and network reliability

## References

- [Wikipedia: Disjoint-set data structure](https://en.wikipedia.org/wiki/Disjoint-set_data_structure)
- [CP-Algorithms: Disjoint Set Union](https://cp-algorithms.com/data_structures/disjoint_set_union.html)

# `t`

```elixir
@type t() :: %Yog.DisjointSet{parents: term(), ranks: term()}
```

Disjoint Set Union (Union-Find) data structure.

Efficiently tracks a partition of elements into disjoint sets.
Uses path compression and union by rank for near-constant time operations.

**Time Complexity:** O(α(n)) amortized per operation, where α is the inverse Ackermann function

# `add`

```elixir
@spec add(t(), term()) :: t()
```

Adds a new element to the disjoint set.

The element starts in its own singleton set.
If the element already exists, the structure is returned unchanged.

## Example

    iex> dsu =
    ...>   Yog.DisjointSet.new()
    ...>   |> Yog.DisjointSet.add(1)
    ...>   |> Yog.DisjointSet.add(2)
    iex> Yog.DisjointSet.size(dsu)
    2

# `connected?`

```elixir
@spec connected?(t(), term(), term()) :: {t(), boolean()}
```

Checks if two elements are in the same set (connected).

Returns the updated disjoint set (due to path compression) and a boolean result.

## Example

    iex> dsu = Yog.DisjointSet.from_pairs([{1, 2}, {3, 4}])
    iex> {_dsu2, result1} = Yog.DisjointSet.connected?(dsu, 1, 2)
    iex> result1
    true
    iex> {_dsu3, result2} = Yog.DisjointSet.connected?(dsu, 1, 3)
    iex> result2
    false

# `count_sets`

```elixir
@spec count_sets(t()) :: non_neg_integer()
```

Returns the number of disjoint sets.

Counts the distinct sets by finding the unique roots.

## Example

    iex> dsu = Yog.DisjointSet.from_pairs([{1, 2}, {3, 4}])
    iex> # 2 sets: {1,2} and {3,4}
    iex> Yog.DisjointSet.count_sets(dsu)
    2

# `find`

```elixir
@spec find(t(), term()) :: {t(), term()}
```

Finds the representative (root) of the set containing the element.

Uses path compression to flatten the tree structure for future queries.
If the element doesn't exist, it's automatically added first.

Returns a tuple of `{updated_disjoint_set, root}`.

## Example

    iex> dsu =
    ...>   Yog.DisjointSet.new()
    ...>   |> Yog.DisjointSet.add(1)
    ...>   |> Yog.DisjointSet.add(2)
    ...>   |> Yog.DisjointSet.union(1, 2)
    ...>   |> Yog.DisjointSet.union(1, 3)
    iex> {_, root} = Yog.DisjointSet.find(dsu, 3)
    iex> root
    1

# `from_pairs`

```elixir
@spec from_pairs([{term(), term()}]) :: t()
```

Creates a disjoint set from a list of pairs to union.

This is a convenience function for building a disjoint set from edge lists
or connection pairs. Perfect for graph problems, AoC, and competitive programming.

## Example

    iex> dsu = Yog.DisjointSet.from_pairs([{1, 2}, {3, 4}, {2, 3}])
    iex> # Results in: {1,2,3,4} as one set
    iex> {_, root1} = Yog.DisjointSet.find(dsu, 1)
    iex> {_, root4} = Yog.DisjointSet.find(dsu, 4)
    iex> root1 == root4
    true

# `new`

```elixir
@spec new() :: t()
```

Creates a new empty disjoint set structure.

## Example

    iex> dsu = Yog.DisjointSet.new()
    iex> Yog.DisjointSet.size(dsu)
    0

# `size`

```elixir
@spec size(t()) :: non_neg_integer()
```

Returns the total number of elements in the structure.

## Example

    iex> dsu =
    ...>   Yog.DisjointSet.new()
    ...>   |> Yog.DisjointSet.add(1)
    ...>   |> Yog.DisjointSet.add(2)
    iex> Yog.DisjointSet.size(dsu)
    2

# `to_lists`

```elixir
@spec to_lists(t()) :: [[term()]]
```

Returns all disjoint sets as a list of lists.

Each inner list contains all members of one set. The order of sets and
elements within sets is unspecified.

Note: This operation doesn't perform path compression, so the structure
is not modified.

## Example

    iex> dsu = Yog.DisjointSet.from_pairs([{1, 2}, {3, 4}, {5, 6}])
    iex> result = Yog.DisjointSet.to_lists(dsu)
    iex> length(result)
    3

# `union`

```elixir
@spec union(t(), term(), term()) :: t()
```

Merges the sets containing the two elements.

Uses union by rank to keep the tree balanced.
If the elements are already in the same set, returns unchanged.

## Example

    iex> dsu =
    ...>   Yog.DisjointSet.new()
    ...>   |> Yog.DisjointSet.add(1)
    ...>   |> Yog.DisjointSet.add(2)
    ...>   |> Yog.DisjointSet.union(1, 2)
    iex> {_, root1} = Yog.DisjointSet.find(dsu, 1)
    iex> {_, root2} = Yog.DisjointSet.find(dsu, 2)
    iex> # Both elements now have the same root
    iex> root1 == root2
    true

---

*Consult [api-reference.md](api-reference.md) for complete listing*